Double Field Theory and strings at the self-dual radius Mariana - PowerPoint PPT Presentation

Double Field Theory and strings at the self-dual radius Mariana Graña CEA / Saclay France In collaboration with arXiv:1509.xxxx G. Aldazabal, S. Iguri, M. Mayo, C. Nuñez, A.Rosabal Mainz, September 2015

momentum # winding # Motivation N − 2) + p 2 p 2 M 2 = 2 R 2 + ˜ α 0 ( N + ¯ ˜ R 2 Bosonic closed string ˜ R = α 0 √ R ≫ l s = l s = α 0 R R l s Massless states: ¯ p = ˜ p = ± 1 N = 0 N = 1 g mn → g µ ν g, B, , dilaton 2 vectors g µy vector 2 scalars g yy scalar ¯ p = − ˜ p = ± 1 N = 1 N = 0 + → B mn B µ ν 2 vectors B µy 2 scalars vector N = ¯ p = ± 2 N = 0 2 scalars 2 scalars p = ± 2 ˜ SU (2) × SU (2) U (1) × U (1) 1 scalar 9 scalars Can we find the effective action using DFT ? ¯ describe the physics using DFT ? N − N = p ˜ p

Some easy math... M = M d × S 1 dof g µ ν B µ ν → d 2 O ( d + 3 , d + 3) → 6 vectors 6 d ⇥ ⇤ = ( d + 3) 2 dim O ( d + 3) × O ( d + 3) 9 scalars → 3 2 ( d + 3) 2

Outline • Strings on S 1 • Effective action from string theory • DFT description • Effective action from DFT • “Internal double space”

String theory on R x = x L + x R Momentum state for non-compact coordinate z ) = x L ( z ) + x R (¯ e ik ( x L ( z )+ x R (¯ X ( z, ¯ z ) z )) k 2 R String theory on S 1 y = y L + y R ' y + 2 π R Momentum state for compact coordinate Momentum & winding state for compact coordinate z ) = y L ( z ) + y R (¯ Y ( z, ¯ z ) k L,R = p ± ˜ e i ( k L y L ( z )+ k R y R (¯ p z )) ˜ ˜ R z ) = y L ( z ) � y R (¯ R Y ( z, ¯ z ) DFT y = y L � y R ' y + 2 π ˜ ˜ R M d ⇥ S 1 M d ⇥ S 1 ⇥ ˜ S 1

N − 2) + p 2 p 2 R 2 + ˜ R = ˜ M 2 = 2( N + ¯ Massless states at R = 1 ˜ R 2 ¯ Level-matching N − N = p ˜ p , ¯ • SU(2) L Vectors α 0 = 1 N x = 1 V ∼ J 3 ( z ) · (¯ A 3 - N y = 1 @ X µ e ikX ) ( g µy + B µy ) : A µ - N y = 0 : A ± p = ˜ p = ± 1 ( k L = ± 2) V ∼ J ± ( z ) · (¯ @ X µ e ikX ) µ A i → ¯ • SU(2) R Vectors A i N x = 1 J i ( z ) → ¯ J i (¯ z ) y L → y R • Scalars (3,3) N x = ¯ N x = 0 (a) N y = 1 , ¯ M 33 ( g yy ) : N y = 1 J 3 ( z ) = ∂ y L ( z ) p = ± 1 (¯ M 3 ± : (b) N y = 1 , p = − ˜ k = ± 2) J ± ( z ) = e ± 2 iy L ( z ) (c) ¯ N y = 1 , p = ˜ p = ± 1 ( k = ± 2) M ± 3 : J i ( z ) = P J i m z − ( m +1) P p = 0 ( k = ¯ M ±± (d) p = ± 2 , ˜ k = ± 2) : n ] = m [ J i m , J j 2 � ij � m, − n + ✏ ijk J k m + n p = ± 2 ( k = − ¯ (e) p = 0 , ˜ k = ± 2) M ± ⌥ : SU(2) L current algebra V ij ⇠ J i J j e ikX

Effective action from string theory Computing 3-point functions <V V V> we read off Gerardo Aldazabal’s talk R − 1 12 H µ νρ H µ νρ + 1 µ ν F iµ ν + 1 ¯ µ ν ¯ 4 F i F i F iµ ν = L 4 +1 F jµ ν + D µ M ij D µ M ij − det M µ ν ¯ M ±± , M ± ⌥ 4 M ij F i acquire mass 2 = ✏ A ± acquire mass 2 H = dB + A i ∧ F i − ¯ A i ∧ ¯ = ✏ 2 F i ¯ A ± F i = dA i + ✏ ijk A j ∧ A k SU(2) x SU(2) → U(1) x U(1) D µ M ii = @ µ M ii + f ijk A j µ M ki + f ijk ¯ A j µ M ik Higgs mechanism M ij → ✏ � ij 33 + M 0 ij

⇒ M d ⇥ S 1 M d ⇥ S 1 ⇥ ˜ Double field theory S 1 In GG/DFT on S 1 y ˜ y ˜ natural pairing R R � T ˜ 1 ⊕ T ⇤ S 1 TS 1 S 1 < @ y + dy, @ y + dy > = 2 ◆ ∂ y dy = 2    0 1 η + dy @ y MN =    < V, V > = η MN V M V N 1 0 ' ' ∂ ˜   η LR = y  1 0 y > = 1 < ∂ y , ∂ ˜  0 − 1 ⇒ to reproduce string theory action Vertex operators: depend on and y R y L we need dependence on and y − = y ˜ = y − ˜ y = y + ˜ y Violating weak / strong constraint ? Yes, as expected: ∂ y ' ∂ ˜ y ¯ Level matching condition N − N = p ˜ p y ( ) = 0 ∂ y ∂ ˜ in usual weak constraint =0 here =0 massless states η MN ∂ M ∂ N ( ) = 0

GG/DFT    e a � ι e a B Frame on T M ⊕ T ∗ M E A =  e a dual e a e a frame frame H ⌦   Contains g, B  g − 1 � g − 1 B Generalized metric H = δ AB E A ⌦ E B H =  O ( D,D ) dof: Bg − 1 g � Bg − 1 B O ( D ) × O ( D )

B µy Circle reduction #   φ � 1 ( ∂ y + B 1 )  e a � ι e a B Frame on T M ⊕ T ∗ M E A =  a e ˆ e a dual e a e a = φ ( dy + V 1 ) g µy frame frame √ g yy = R T M = T M d ⊕ TS 1 y ∼ y + 2 π

Circle reduction B µy #   φ � 1 ( ∂ y + B 1 )  e a � ι e a B Frame on T M ⊕ T ∗ M E A =  = φ ( dy + V 1 ) g µy e a dual frame e a frame e a a e ˆ √ g yy = R = exp( 1 2 M 33 ) . � = e < > T M = T M d ⊕ TS 1 2 < M 33 > ≈ 1 + 1 y ∼ y + 2 π ✏                 L  U +  E U − 1 2 M 33  φ � 1 1 0  ∂ y + B 1  E d  J + A  = = =          R 2 M 33 1 U + J − ¯ ¯ E U − 1 E d 0 dy + V 1 η LR φ A Scherk-Schwarz A 0 ( x ) E E A ( x, y ) = ) = U A x ) E 0 A 0 ( y ) reduction U ± = 1 U + ≈ 1 2( φ − 1 ± φ ) J = @ y + dy A = V 1 + B 1 U � ≈ 1 2 M 33 ¯ ¯ = A V 1 − B 1 J = @ y − dy , Effective action valid at energies 1 1 E ⇠ α 0 ✏ << p p α 0 So far, no enhancement of symmetry, no double field theory

DFT & Enhancement of symmetry DFT 1 ⊕ T ⇤ M d ⊕ T ˜ = T M d ⊕ TS 1 ⊕ T ⇤ S 1 S 1 T M ⊕ T ∗ M ' dy ' ∂ ˜ y J = @ y + dy = ∂ y + ∂ ˜ = ∂ y L y ¯ J = @ y − dy , = ∂ y � ∂ ˜ = ∂ y R y Still, this is formal. No dependence on y or ˜ y Of course, we have not included momentum/winding modes ⇠ e 2 iy /e 2 i ˜ y S 1 , ˜ S 1 To include winding modes we need DFT: To account for the enhancement of symmetry, we need to enlarge the generalized tangent space 1 ⊕ T ⇤ M d ⊕ T ˜ = T M d ⊕ TS 1 S 1 T M d ⊕ V 2 ⊕ TS 1 ⊕ T ˜ S 1 ⊕ V ∗ 2 ⊕ T ∗ M d O(3,3)

      Enhancement of symmetry                J 3 + A 3 L 2 M 3¯ 1  E 3 3 1 1  E 2 M 33  J + A 1  = 1 ⊕ T ⇤ M d  = ⊕ T ˜ = T M d ⊕ TS 1 S 1         ¯ 3 − ¯ E ¯ 2 M ¯ J ¯ A ¯ J − ¯ ¯ R 3 2 M 33 1 1 33 3 1 1 E A             T M d ⊕ V 2 ⊕ TS 1 ⊕ T ˜ S 1 ⊕ V ∗        J j + A j 2 ⊕ T ∗ M d  E i 2 M i ¯ 1 1 |  =    ¯ | − ¯ ı 1 ıj E ¯ 2 M ¯ J ¯ A ¯ 1 | M i ¯ A i ( x ) | ( x ) ¯ ı ( x ) A ¯ 9 scalar fields 6 vector fields J i ( y, ˜ Should satisfy SU(2) L algebra y ) ¯ ı ( y, ˜ Should satisfy SU(2) R algebra J ¯ y ) under some bracket

Effective action       ◆ e a ¯ 1 0 0 0 E a e a ◆ e a A A ◆ e a B       M ¯ L 0 1 1 0 0 0  E   M A   J  2       =       ¯ R M t M t A 0 1 0 0 0  E   1   J     2          E a e a 0 0 0 0 0 0 1 A 0 ( x ) E E A ( x, y ) = ) = U A x ) E 0 A 0 ( y ) Generalized Scherk-Schwarz reduction of DFT action Aldazabal, Baron, Marques, Nuñez 11 Geissbuhler 11 0 1 2 @ 1 M ≈ I = i, ¯ ı A M t 1 R − 1 12 H µ νρ H µ νρ + 1 R − 1 12 H µ νρ H µ νρ + 1 µ ν F iµ ν + 1 F iµ ν + 1 F jµ ν + D µ M ij D µ M ij ¯ µ ν ¯ µ ν ¯ 4 H IJ F Iµ ν F J µ ν + ( D µ H ) IJ ( D µ H ) IJ 4 F i F i 4 M ij F i = L = L 4 − 1 H IL H JM H KN − 3 H IL η JM η KN + 2 η IL η JM η KN � Exactly string theory action! � 12 f IJK f LMN − det M K ] = f IJK E 0 dB + F I ∧ A I [ E 0 J , E 0 H = K C dA I + f I JK A J ∧ A K J, ¯ F I = J ✏ ijk , ✏ ı | k

Algebra + V 2 + TS 1 + T ˜ V 2 + TS 1 + TS 1 + V 2 S 1 + V ∗ 2 + C-bracket v L 1 , v L v R 1 , v R 2 2 [ V 1 , V 2 ] C = 1 2( L V 1 V 2 − L V 2 V 1 ) ( ( L V 1 V 2 ) I = V J 1 ∂ J V I 2 + ( ∂ I V 1 J � ∂ J V I 1 ) V J generalized Lie derivative 2    ¯ The following and do the job      J + J −         cos 2 y R sin 2 y R v R 0 cos 2 y L sin 2 y L v L 0 1 1         ¯ J = − sin 2 y R cos 2 y R v R 0     J = − sin 2 y L cos 2 y L v L 0     2     2             dy R 0 0 1 dy L 0 0 1        K ] = f IJK E 0 [ E 0 J , E 0 K C J, ¯ J ✏ ijk , ✏ ı | k